permutations problem - student may not win all 23 prizes

Problem is:
3 prizes, one for English, one for French and one for Spanish can be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if:
(i) no student may win > 1 prize.
(ii) no student may win all 3 prizes.

3 prizes, one for English, one for French and one for Spanish can be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if:
(ii) no student may win all 3 prizes.
My first thought was that way to do this is 20*20*19 - but that is incorrect. Correct way to solve is apparently 20*20*20 - 20. But why?

There are functions from a set of three to a set of twenty. But there are 20 ways that all three go to the same image (in this case a student). So subtract..